If we interpret these numbers as bases and exponents, we can say (using the diamond pattern) that one is functioning as an exponent for the number four and as a base for the number two when moving in a clockwise direction.

How important this is to math I don't know nor do I know if someone else ever investigated this. To me it looked interesting enough to list which may stimulate thought.

If we interpret these numbers as bases and exponents, we can say (using the diamond pattern) that one is functioning as an exponent for the number four and as a base for the number two when moving in a clockwise direction.

How important this is to math I don't know nor do I know if someone else ever investigated this. To me it looked interesting enough to list which may stimulate thought.

PhilX

PS There is another meaning, more common, for the term cyclic number.

I have observed similarities to what you are observing. If viewed as a cyclic function the number one would have to exists through a "self-reflection" process where it is spatially equivalent to a 1 dimensional point reflecting into itself:

As all number is composed upon a self-reflecting one, all 1n follows the same form and function

(1,2,-1) ≡ (1,2,-1) → (-2,-1,0,1,2,3,4)

*****With all positive numbers manifesting at double the rate of the negative.

a) 1 ≡ 2 → 3

b) 1 ≡ -1 → 0

c) 2 ≡ 2 → 4

d) 2 ≡ -1 → 1

e) -1 ≡ -1 → -2

The question I have is which came first the form (1,2,3,etc.) or the function (+,-, etc.)?

We can observe that:

Through self reflection as "cycling":

1) addition cycling (reflecting) upon itself results in multiplication as the addition of addition. This is considering that addition may be viewed strictly as a structural extension of 1 as a Positive.

2) multiplication cycling upon itself results in exponention.

3) subtraction cycling upon itself results in division as the subtraction of subtraction. This is considering that subtraction may be viewed strictly as a structural extension of 1 as a Negative.

4) division cycling upon itself results in roots.

Through dualities as "rotations":

⟨+|-⟩ → +1- → ⟨+|-⟩

****with "1" being the axis from which it begins as arithmetic and subtraction are the first degrees of numerical "curvature".
****Addition and subtraction exist if and only if there is "1".
****Addition and subtraction can be viewed as equivalent to the reflection of a Positively valued 1 or a Negatively valued 1.

⟨*|/⟩ → x2/ {or (+1-) ≡ (+1-)} → ⟨*|/⟩

**** with "2" being the axis from which it begins as multiplication and division are the second degrees of numerical "curvature".
**** Multiplication and Division exist if and only if there is "1 reflecting upon itself (cycling)" as 2. 2 dividing itself results in the 1 as a further foundation for standard multiplication and division.

Considering that all cycles seem to stem from Pi in theory their should be some cyclic connection as a result of 1 reflecting upon itself. I am still working on that aspect. The Pythagoreans had a theory where "3" was actually the "first" number and it came before 1 or at the same time as 1. If this holds true than in theory Pi may hold a key to the origins of mathematics and the nature of spatial properties.

If number is viewed as a spatial element conducive to a "point" and nothing more Pi results in an infinite particle wave function that in theory would hold all real world potential "realities" (as all physical realities are composed of particle waves).

These are the complex n-th roots of 1. As an example, consider the integers mod 4. These are the numbers 0, 1, 2, 3 with addition mod 4, meaning that the addition cycles around. So 2 + 2 = 0, 2 + 3 = 1, and so forth.

In the complex numbers, with i defined as a number such that i^2 = -1, we can look at the set of integer powers of i. We have:

i = i

i^2 = -1

i^3 = -i

i^4 = 1

and then these powers continue cycling in that pattern.

Geometrically, we are taking a vector pointing east and repeatedly rotating it a quarter turn to the left. East, North, West, South, and then back to east. These are the powers of i. These are the integers mod 4. These are the quarter rotations of the plane.

You can do the same trick for the 5-th roots of 1, or in general the n-th roots. The n-th complex roots of 1 are a representation of the integers mod n. They represent a rotation of the plane through an angle of 2pi/n.

If you are interested in the relationship among pi, rotations, and cyclic groups of integers, take a look at the Wiki page I linked.

These are the complex n-th roots of 1. As an example, consider the integers mod 4. These are the numbers 0, 1, 2, 3 with addition mod 4, meaning that the addition cycles around. So 2 + 2 = 0, 2 + 3 = 1, and so forth.

In the complex numbers, with i defined as a number such that i^2 = -1, we can look at the set of integer powers of i. We have:

i = i

i^2 = -1

i^3 = -i

i^4 = 1

and then these powers continue cycling in that pattern.

Geometrically, we are taking a vector pointing east and repeatedly rotating it a quarter turn to the left. East, North, West, South, and then back to east. These are the powers of i. These are the integers mod 4. These are the quarter rotations of the plane.

You can do the same trick for the 5-th roots of 1, or in general the n-th roots. The n-th complex roots of 1 are a representation of the integers mod n. They represent a rotation of the plane through an angle of 2pi/n.

If you are interested in the relationship among pi, rotations, and cyclic groups of integers, take a look at the Wiki page I linked.

Thanks, that is actually a great insight I will look into when I have the time. I argued on a separate thread that axioms have a similar format to imaginary numbers, but the thought never occured to me to use "i" for this "setup".

In simpler terms, what I implied through the above post, is that all number is strictly founded upon 1 curving upon itself through a intra-dimensional cycle.

This cycle results in one having both positive and negative values, with these positive and negative values being equivalent to the addition and subtraction we observe as the foundations of arithmatic. This implies that number and arithmatic are inseperable entities and exist fundamentally as "reflective" values.

I'm making an effort to try to relate your ideas to something in standard math that makes sense to me.

Now, 1 + 3 = 4 I understand.

Can you define your use of the symbols ≡ and ≅?

When you say +1 ≡ +3 ≅ +4 what do you mean by that? Can you explain it to someone not familiar with your private notation? It's perfectly legal for you to define your symbols any way you like. But you have to say how you are defining them otherwise nobody can understand you.

I'm making an effort to try to relate your ideas to something in standard math that makes sense to me.

Now, 1 + 3 = 4 I understand.

Can you define your use of the symbols ≡ and ≅?
≡ = "reflect" with "reflect" meaning to "direct while maintaining integral structure"
≅ = "congruent in structure"

When you say +1 ≡ +3 ≅ +4 what do you mean by that? All positive and negative values are duels whose axis is "1". 1 as an intradimensional structure maintains itself through reflection while simultaneously manifesting further numbers as structural extensions of itself (considering all number can be considered one reflecting upon itself). In manifesting further numbers 1 as a positive value manifests negative values simultaneously.

What we understand of positive and negative values breaks down to "addition" and "subtraction". In these respect when a positive one reflects a positive three it is equivalent to saying one plus three. The same applies for subtraction, or the addition of negatives.

As one is infinite in self reflection, 4 as a structure of one self-reflecting, always exists so positive one reflecting positive three "is congruent in structure" to four.

I have a whole "equation" arguing for one being equal to infinity on a separate thread.

In these respect one as an intradimensional structure manifests further structures (numbers) through self reflection. Considering it simultaneously manifests addition and subtraction, through positive and negative values, when it further "reflects" it simultaneously manifests multiplication (as the addition of addition) and division (as the subtraction of subtraction).

Can you explain it to someone not familiar with your private notation? It's perfectly legal for you to define your symbols any way you like. But you have to say how you are defining them otherwise nobody can understand you.

These are the complex n-th roots of 1. As an example, consider the integers mod 4. These are the numbers 0, 1, 2, 3 with addition mod 4, meaning that the addition cycles around. So 2 + 2 = 0, 2 + 3 = 1, and so forth.

In the complex numbers, with i defined as a number such that i^2 = -1, we can look at the set of integer powers of i. We have:

i = i

i^2 = -1

i^3 = -i

i^4 = 1

and then these powers continue cycling in that pattern.

Geometrically, we are taking a vector pointing east and repeatedly rotating it a quarter turn to the left. East, North, West, South, and then back to east. These are the powers of i. These are the integers mod 4. These are the quarter rotations of the plane.

You can do the same trick for the 5-th roots of 1, or in general the n-th roots. The n-th complex roots of 1 are a representation of the integers mod n. They represent a rotation of the plane through an angle of 2pi/n.

If you are interested in the relationship among pi, rotations, and cyclic groups of integers, take a look at the Wiki page I linked.

If you said these are your own private ideas and notation and that you have a hard time explaining them to people, that would be believable.

Your ideas and notation are not standard math and don't correspond to anything in standard math.

In the context of the integers, the symbol ≡ means "congruent mod n".

For example 5 ≡ 3 (mod 2) because 5 and 3 have the same remainder when divided by 2

Likewise 9 ≡ 16 (mod 7) because 9 and 16 have the same remainder when divided by 7.

But your use of the symbol ≡ is nonstandard and you haven't explained it in a way that makes sense.

If you think your exposition and notation are public knowledge you're wrong about that. Your use of the word "reflection" is nonstandard and you haven't explained it. Your formula +1 ≡ +3 ≅ +4 has not been explained to my satisfaction.

It's true that 1 ≡ 3 (mod 2) but that's as much sense as I can make of this. It's also true that 2 ≡ 4 (mod 2) but your usage of the symbol ≅ is unexplained.

You haven't defined it or said what it means. The nearest I can figure is that by reflection you mean that whole numbers come in +/- pairs, so that given 1 we also have -1. The usual terminology is that 1 and -1 are the additive inverses of each other.

You haven't defined it or said what it means. The act of direction, or "the throwing back by a body or surface of light, heat, or sound without absorbing it" (bing)
The nearest I can figure is that by reflection you mean that whole numbers come in +/- pairs, so that given 1 we also have -1. The usual terminology is that 1 and -1 are the additive inverses of each other.

I am not arguing that whole numbers come in pairs but rather +/- are extensions of the numbers themselve and +/- cannot be seperate from 1 as 1 is +1- in nature. To argue that +1 and -1 exist as seperate entities is to argue that 1 is the axis of the duality of +/-.

1 "reflects" or "directs itself into itself". In doing this a cycle is achieved through reflection which in turn propogates further numbers as extensions of itself.

That's just word salad.

But one question occurs to me. Are you doing math, metaphysics, or poetry? Perhaps I'm taking you too literally and trying to make mathematical sense of your ideas. Perhaps you are not trying to make mathematical sense, but are perhaps trying to explain the nature of numbers in the universe or something along those lines. In which case I still can't understand you, but I would agree that it's not helpful for me to ask you to make mathematical sense.

1 "reflects" or "directs itself into itself". In doing this a cycle is achieved through reflection which in turn propogates further numbers as extensions of itself.

That's just word salad.

But one question occurs to me. Are you doing math, metaphysics, or poetry? Perhaps I'm taking you too literally and trying to make mathematical sense of your ideas.

Try seperating math from the nature of the "axiom" as "self-evidence" and you will get your answer. Most modern math is just word-salad.

Perhaps you are not trying to make mathematical sense, but are perhaps trying to explain the nature of numbers in the universe or something along those lines.
In which case I still can't understand you, but I would agree that it's not helpful for me to ask you to make mathematical sense.

If mathematics can be "sensed" as you suppose, than it has a subjective nature. Strictly speaking the axioms of modern math don't make sense.

1 "reflects" or "directs itself into itself". In doing this a cycle is achieved through reflection which in turn propogates further numbers as extensions of itself.

That's just word salad.

But one question occurs to me. Are you doing math, metaphysics, or poetry? Perhaps I'm taking you too literally and trying to make mathematical sense of your ideas.

Try seperating math from the nature of the "axiom" as "self-evidence" and you will get your answer. Most modern math is just word-salad.

Perhaps you are not trying to make mathematical sense, but are perhaps trying to explain the nature of numbers in the universe or something along those lines.
In which case I still can't understand you, but I would agree that it's not helpful for me to ask you to make mathematical sense.

If mathematics can be "sensed" as you suppose, than it has a subjective nature. Strictly speaking the axioms of modern math don't make sense.